Reconstruction of the Initial Data and Source Functions for a Wave Equation with Nonlocal Boundary Condition

This paper addresses multiple inverse source problems linked to the wave equation under nonlocal boundary conditions. A bi-orthogonal functional framework is adopted to represent the solutions through series expansions. The analysis establishes that these problems are ill-posed in the Hadamard sense. Three main reconstruction tasks are considered: identification of an unknown space-dependent source, recovery of the initial data, and estimation of a time-varying source term. For each case, suitable additional conditions are introduced to ensure the uniqueness of the unknown quantities. Existence and uniqueness theorems are proved under specific smoothness requirements. Finally, the theoretical developments are validated through numerical computations.